Probability Asymptotic Theorems and Their Applications

本研究计划致力于与概率和统计中的一般渐近定理及其在各种问题中的应用相关的主题。概率渐近理论是概率论和统计学的核心，在几乎所有的科学或社会科学分支中起着中心作用，包括天气预报、地震学等。诸如大数定律、中心极限定理和独立随机变量的重对数定律等结果形成了现代概率论。它们已经在许多方向上扩展和推广，其中包括更一般的随机过程和随机测度，并且它们已经成为渐近统计或大样本理论的基础，这是评估估计量和统计检验属性的通用框架。在这个框架内，通常假设样本量n无限增长，并且当样本量n趋于无穷大时，在极限中评估统计程序的属性。 这个研究建议的第一个重点涉及到我我长期以来的研究兴趣几乎肯定和弱收敛的随机过程，特别是在法律的大数，中心极限定理，法律的重对数，概率大和中度偏差，并精确渐近的经典极限定理的实值或Banach空间值随机过程。目的是研究经典极限结果的改进，发展一些新的方法，建立一些新的概率不等式来证明随机过程的几乎处处收敛和弱收敛。我还打算继续我以前的研究，使用现代随机过程技术的概率，找到条件下，几乎肯定和弱收敛举行随机过程，并调查统计应用这种收敛。 本研究计划的第二个重点将是调查统计应用中的渐近行为，涉及高维数据分析问题，分层模型，回归方法，随机矩阵理论等。因此，我建议研究a）自举样本相关矩阵的最大项，B）球形系综矩阵乘积的谱半径，c）分层模型，等 成功地完成我提出的工作将是一个重要的一步，增加我们的理解的渐近行为的随机过程中非常普遍和适用的情况下，因为结论的渐近定理往往补充的结论，可以通过数值方法。在这个建议中得到的结果将是新颖的和有意义的，因为它们将扩展，概括，并完善文献中的早期工作。所有研究结果将以论文形式正式化，提交给高影响力的学术期刊发表。

This research proposal is devoted to topics related to general asymptotic theorems in probability and statistics, and their applications to a wide variety of problems. Probability asymptotic theory lies at the heart of probability and statistics, and plays a central stage in almost every branch of science or social science including weather forecasting, psephology, etc. Results such as the law of large numbers, the central limit theorem, and the law of the iterated logarithm for independent random variables have given shape to modern probability theory. They have been extended and generalized in many directions, among others, to more general random processes and random measures, and they have become the basis of asymptotic statistics or large sample theory which is a generic framework for the assessment of properties of estimators and statistical tests. Within this framework, it is typically assumed that the sample size n grows indefinitely, and the properties of statistical procedures are evaluated in the limit as the sample size n tends to infinity. The first focus of this research proposal relates to my my long-standing research interest in almost sure and weak convergence of random processes, especially in the laws of large numbers, central limit theorems, the law of the iterated logarithm, probabilities of large and moderate deviations, and precise asymptotics in the classical limit theorems for real-valued or Banach space-valued random processes. The goal is to study refinements of the classical limit results, to develop some new methods and to establish some new probability inequalities for proving almost sure and weak convergence of random processes. I also aim to continue my previous research, using modern random process techniques in probability to find conditions under which almost sure and weak convergence holds for random processes, and to investigate statistical applications of such convergence. A second focus of this research proposal will be on investigating the asymptotic behaviour in statistical applications pertaining to high dimensional data analysis problems, hierarchical models, resampling methods, random matrix theory, etc. Therefore I propose to study the asymptotic theorems for a) largest entries of bootstrapped sample correlation matrices, b) spectral radii for product of matrices from the spherical ensemble, c) hierarchical models, etc. The successful completion of my proposed work would be an important step in increasing our understanding of the asymptotic behaviour of random processes in very general and applicable situations since the conclusions of an asymptotic theorem often supplement the conclusions which can be obtained by numerical methods. The results to be obtained in this proposal will be novel and significant in that they will extend, generalize, and refine earlier work in the literature. All results will be formalized in papers submitted for publication in high-impact academic journals.